What will it cost to put right?
Why did you get it checked?
What are the specific grounds of the complaint?
What exactly has he got wrong?
Is the new roofer willing to go to court and give this evidence if required?
Yes, but it is not that simple. That is the sort of thing that lay people say without really considering the meaning. It is an emotive statement which isn't sufficient.
To claim from the original roofer you have to show that the work fell below the standard required. It is not as simple as saying that you are displeased unless you agreed that satisfaction was guaranteed which is not likely with a contract of this value.
Therefore, another roofer needs to specify the ways in which this work is faulty. You would probably be advised to get the opinions of two roofers and quotes from both. Choose the cheapest and write to him with the quote warning him that if he doesn't resolve this you will instruct another and sue for the cost. Give him a specified time period.
If he still doesn't respond then do the work and issue against him at the small claims court.
Can I clarify anything for you?
There isn't any other way to recover than to sue I'm afraid.
That is the only remedy in law.